In Section 12.2 of [1]}, Lusztig introduced a geometric pairing \(\lbrace A,B\rbrace \)  for \(A,B\in {\mathcal {D}}_{G_{{\mathbf {V}}}}^{b,ss}(E_{{\mathbf {V}}})\) , where \({\mathbf {V}}\)  is a fixed \(I\) -graded vector space with dimension vector \(\nu \in \mathbb {N}I\) .
